A Parameterized Perspective on Protecting Elections

We study the parameterized complexity of the optimal defense and optimal attack problems in voting. In both the problems, the input is a set of voter groups (every voter group is a set of votes) and two integers $k_a$ and $k_d$ corresponding to respectively the number of voter groups the attacker can attack and the number of voter groups the defender can defend. A voter group gets removed from the election if it is attacked but not defended. In the optimal defense problem, we want to know if it is possible for the defender to commit to a strategy of defending at most $k_d$ voter groups such that, no matter which $k_a$ voter groups the attacker attacks, the outcome of the election does not change. In the optimal attack problem, we want to know if it is possible for the attacker to commit to a strategy of attacking $k_a$ voter groups such that, no matter which $k_d$ voter groups the defender defends, the outcome of the election is always different from the original (without any attack) one.

💡 Research Summary

This paper investigates the computational complexity of two fundamental security problems in elections when both an attacker and a defender have limited resources. The setting consists of a collection of voter groups—each group being a multiset of identical votes—and two integers, kₐ (the maximum number of groups the attacker may delete) and k_d (the maximum number of groups the defender may protect). A group is removed only if it is attacked and not defended.

The Optimal Defense problem asks whether the defender can choose at most k_d groups to protect such that, regardless of which kₐ groups the attacker subsequently deletes, the election’s winner(s) remain exactly as in the original, unperturbed election. The Optimal Attack problem asks whether the attacker can select at most kₐ groups so that, no matter which k_d groups the defender later protects, the election outcome is guaranteed to differ from the original.

The authors study these problems for all normalized scoring rules (including Plurality, Borda, k‑approval, etc.) and for the Condorcet rule. Their first major contribution is to prove that both problems are NP‑hard even when the number of candidates m is as small as three. This strengthens earlier results that only considered larger candidate sets.

In the parameterized complexity framework, they consider several natural parameters: the attacker’s budget kₐ, the defender’s budget k_d, and the combined pair (kₐ, k_d). They obtain a detailed hardness landscape:

• Optimal Defense is W